Document Type: Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.


The aim of this paper is to obtain three weak solutions for the Dirichlet quasilinear elliptic systems on a bonded domain. Our technical approach is based on the general three critical points theorem obtained by Ricceri.


Main Subjects

[1] E. Acerbi and G. Mingione,  Regularity results for stationary electro-rheological fluids,  Arch. Ration. Mech. Anal.,  156 (2001), pp. 121-140.

[2] X.L. Fan, J. Shen, and D. Zhao,  Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl., 262 (2001), pp. 749-760.

[3] X.L. Fan and D. Zhao,  On the spaces Lp(x)(Ω) and W1,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), pp. 424-446.

[4] X.L. Fan and Q.H. Zhang,  Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), pp. 1843-1852.

[5] A.El. Hamidi,  Existence result to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl., 300 (2004), pp. 30-42.

[6] J.J. Liu and X.Y. Shi,  Existence of three solutions for a class of quasilinear elliptic systems involving the (p(x), q(x))-Laplacian, Nonlinear Anal., 71 (2009), pp. 550-557.

[7] M. Mihailescue,  Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 2007 (67), pp. 1419-1425.

[8] B. Ricceri,  A three critical points theorem revisited, Nonlinear Anal., 70 (2009), pp. 3084-3089.

[9] B. Ricceri,  A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), pp. 401-410.

[10] B. Ricceri,  Existence of three solutions for a class of elliptic eigenvalue problem, Math. Comput. Modelling., 32 (2000), pp. 1485-1494.

[11] M. Ruzicka,  Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math, vol. 1784, Springer-Verlag, Berlin, 2000.

[12] H.H. Yin,  Existence of three solutions for a Neumann problem involving the p(x)-Laplace operator, Math. Meth. Appl. Sci., 35 (2012), pp. 307-313.

[13] V.V. Zhikov,  Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), pp. 33-36.